The wave maps equation is perhaps the simplest incarnation of a geometric wave equation -- the nonlinearity arises naturally from the Riemannian structure of the target manifold. It is the hyperbolic analogue of the (elliptic) harmonic maps equation and the (parabolic) harmonic map heat flow.

We will review some of the significant developments from the past decade concerning the asymptotic dynamics of solutions to the energy critical wave maps equation, emphasizing the crucial role that harmonic maps play in singularity formation. We will also discuss new phenomena that arise when curvature is introduced in the domain, focusing on the theory of wave maps on hyperbolic space that is being developed jointly with Sung-Jin Oh and Sohrab Shahshahani.